Two dimensional zonoids and Chebyshev measures
type: Item
instance of: scholarly article
title: Two dimensional zonoids and Chebyshev measures
zbMATH Open document ID: 0885.28007
author: Stefano Bianchini
author: Raphaël Cerf
author: Carlo Mariconda
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-04-01T00:00:00Z
review text: An \(\mathbb{R}^2\)-valued measure \(\mu= (\mu_1,\mu_2)\) on \([0,1]\) (with a \(\sigma\)-field) is a Chebyshev measure if it is non-atomic and each \(|\mu|^{\otimes 2}\)-non-negligible measurable subset of \([0,1]\times[0, 1]\) has positive \(\text{det }\mu\) measure, where \(\text{det }\mu= \mu_1\otimes\mu_2- \mu_2\otimes\mu_1\). The authors prove that a set \(K\subset\mathbb{R}^2\) is the range of some Chebyshev measure if and only if \(K\) is strictly convex, compact, centrally symmetric, and contains \((0,0)\) in its boundary. Further results for \(\mathbb{R}^2\)-valued measures \(\mu\) are: a condition under which \(\mu\) is the difference of two Chebyshev measures, a condition on the density of \(\mu\) (with respect to its variation measure \(|\mu|\)) that is necessary for the strict convexity of the range of \(\mu\), and a characterization of the Chebyshev measures.
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 1053436
zbMATH Keywords: range of a vector measure
zbMATH Keywords: Lyapunov theorem
zbMATH Keywords: zonoid
zbMATH Keywords: Chebyshev measure
MaRDI profile type: Publication
OpenAlex ID: W2072488207
cites work: A Class of Convex Bodies
cites work: On Bang–Bang Constrained Solutions of a Control System
cites work: Oriented measures with continuous densities and the bang-bang principle
cites work: Oriented measures
cites work: Chebyshev measures
cites work: Q5643279
cites work: Q4838489
cites work: Decomposition of ranges of vector measures
cites work: Q5599348
cites work: Q4340096
cites work: Q4135799
DOI: 10.1006/JMAA.1997.5486
description: scientific article; zbMATH DE number 1053436